3 Answers
3

This is not an official Star Trek answer, but it is too long for a comment.

The reason not all personnel are armed is due to several "real world" reasons.
First, I am unsure if all ST personnel are required to take fire arms qualifications during their training, but in today's military they are... but I would not trust a good chunk of them with a weapon, given their primary occupation. While they may have used the weapons at some point and been trained, they would have "lost their edge" after spending time learning/preparing for their "actual" job and even less time after they began their "actual" job. This would result in a decreased reaction time using the weapons and possibly even forgetting how to handle weapons cleaning, malfunctions, etc.

Tagging off of this first fact is the point of the actual teams dedicated to ship security. The security teams did have weapons training are assuredly running training and tests everyday to prepare themselves in the event the ship is boarded. They are trained, work well as a team, and are ready to carry out specific defensive measures that are aligned to a specific purpose (i.e. coordinating ship sweeps that push the intruders to a corner or non-essential area). Working in the galley, engineering, etc. every day would not allow the rest of the crew to participate in those training sessions. This would cause them to be unreliable and potentially a liability should the situation actually happen.

Without the training and comfort of handling weaponry everyday there is also the higher risk of what is called "blue on blue" fire; this is when you fire on your own people... I have had it happen to me and it is scary as hell. Even people who go hunting end up in the emergency room because they shot their buddy; it is a higher risk when you know someone else is gonna shoot back at you and you are scared for your life.

There are potentially other reasons for this, but these will hopefully get the point across. It is the same concept as why not give everyone a gun in America to defend against active shooters? The only way to stop a bad guy with a gun is a good guy with a gun right? we hear that all the time... and while that works in theory, in reality it would not work that way.

Edit: I realized I completely forgot to acknowledge your second question. This is much more straight forward however. The reason they did not consider themselves a military vessel is because they were not a military organization. Star Fleet was about exploration, nothing more. It also bodes well for future contact if you say you are an explorer as opposed to a military might. It takes away some of the tension, not all but some. And as will all explorations you still can bring guns with you when you explore. It is a mindset that you want your organization to have for both themselves and that they present to anyone else you meet. You are not there to cause trouble, but you will defend yourself if provoked.

I believe the previous 'real world' answer on arming people who have other specialties on board is plausible. There is friction between the existing tactical officer, Malcolm Reed, and the additional MACO military contingent put on board for the pursuit of the Xindi. Their issues with one another require intervention by the Captain who has more pressing things on his mind than his own armed crew men's dispute with each other.

To answer your second question about why stick to the idea of exploration you must consider the original vision of the Star Trek universe's creator, Gene Roddenberry, who saw an optimistic future where humanity is not seeking to conquer but to learn. Captain Archer exhibits this ideal strongly in the first season.

Here is a quote from Roddenberry

"It speaks to some basic Human needs, that there is a tomorrow – it's
not all going to be over in a big flash and a bomb, that the Human
race is improving, that we have things to be proud of as humans. No,
ancient astronauts did not build the pyramids – Human beings built
them because they're clever and they work hard. And Star Trek is about
those things." - Eugene Wesley Roddenberry, from the Star Trek 25th Anniversary special, 1991 (as cited by Susan Sackett, used with permission)

If you are still curious about some of the production aspects of Enterprise then look at Memory Alpha.

Generally speaking, you're on board a friendly ship. Why would you walk around with a gun at all times on your own ship? Especially within a confined starship, it would contribute to paranoia.

Granted that this is well after Enterprise, but Star Trek VI kinda touches on this (and there's no reason to think it wasn't true in Archer's day). Firing a weapon without authorization would set off an alarm, summoning security. It could be a constant hassle to have everyone armed all the time

CHEKOV: Why not simply vaporise them?VALERIS: Like this? [fires phaser]KLAXON: Wails. [Security team responds]VALERIS: At ease. As you know, Commander Chekov, no one can fire an unauthorised phaser aboard a starship.snipUHURA: [Entering] Did someone fire a phaser?CHEKOV: It's all right. It's nothing.